Expected Inflation and Other Determinants of Treasury Yields Forthcoming, Journal of Finance

Transcription

1 Expected Inflation and Other Determinants of Treasury Yields Forthcoming, Journal of Finance Gregory R. Duffee Johns Hopkins University Prepared December 2017 Note: This is the version sent to the JF copy editor. It does not reflect the copy editor s comments. The published version will differ slightly from this version.

2 Expected Inflation and Other Determinants of Treasury Yields Gregory R. Duffee ABSTRACT Shocks to nominal bond yields are comprised of news about expected future inflation, news about expected future real short rates, and news about expected excess returns all over the life of the bond. I estimate the magnitude of the first component for short and long maturity Treasury bonds. At a quarterly frequency, variances of news about expected inflation account for between 10 to 20% of variances of yield shocks. Standard dynamic models with long run risk imply corresponding variance ratios close to one. Habit formation models fare somewhat better. The magnitudes of shocks to real rates and expected excess returns cannot be determined reliably in the data owing to statistical uncertainty of the persistence of real rate shocks. Johns Hopkins University. Thanks to seminar participants at many schools and conferences, Ravi Bansal, Mike Chernov, Anna Cieslak, George Constantinides, Lars Lochstoer, Kenneth Singleton (Editor), Jonathan Wright, and three anonymous referees for helpful comments. Thanks especially to discussants Anh Le and Scott Joslin. I have read the Journal of Finance s disclosure policy and have no conflicts of interest to disclose. 1

3 A large and expanding literature explores the relation between nominal bond yields and inflation. Ang and Piazzesi (2003) make a particularly important contribution, introducing Gaussian macro finance dynamic term structure models to determine the compensation investors require to face shocks to inflation and macroeconomic activity. The related literature quickly branched out to include unspanned macro risks, non Gaussian dynamics, and fundamental explanations for inflation risk premia that are grounded in investor preferences and New Keynesian macro models. It is difficult to uncover from this literature any widely accepted conclusions about the joint dynamics of inflation and the nominal term structure. Ang, Bekaert, and Wei (2008) make the same point to motivate their own attempt to produce some basic facts. Research subsequent to Ang, Bekaert, and Wei does not converge on their conclusions or any other set of core results. Thus it is not clear what branches of the macro finance literature are likely to be fruitful and which should be abandoned. In this research I make yet another effort to identify a robust empirical property that can be used to guide future research. How large are shocks to expected inflation relative to shocks to nominal bond yields? Embedded in the question is an accounting identity. Campbell and Ammer (1993) show that the shock to a nominal bond s yield equals the sum of news about expected inflation, news about expected short term real rates, and news about expected excess returns, all over the life of the bond. The inflation variance ratio, as defined here, is the variance of expected inflation news relative to the variance of the yield shock. Three observations motivate a focus on this ratio. First, it can be estimated without much structure, using survey forecasts of inflation to identify revisions in inflation expectations. Second, ratios based on quarterly U.S. data are reliable, in the sense that standard errors are tight and estimates are reasonably stable over time. Third, ratios inferred from these 2

4 data are strongly at odds with corresponding values from both endowment economy long run risk models and standard New Keynesian dynamic models. Results for the ten year horizon are representative. During the past 35 years, the standard deviation of quarterly shocks to ten year average expected inflation is in the neighborhood of 20 basis points. The standard deviation of quarterly shocks to the ten year bond yield is much larger around 60 basis points. Squaring and dividing produces a variance ratio estimate close to 10%. The challenge to economists is easier to see when viewing the result from a different angle. In the data, shocks to nominal yields are large, and are driven primarily by a combination of news about expected short term real rates and news about expected excess returns. In our benchmark macro finance models, channels for both types of news are small. Bansal and Yaron (2004) develop one of these benchmark models, combining a representative agent, recursive utility preferences, and persistent fluctuations in the endowment growth rate. Short term real rates are driven largely by fluctuations in expected consumption growth. The long run risk literature follows Bansal and Yaron by relying on a high elasticity of intertemporal substitution and fairly small shocks to expected growth. The combination of these properties results in low volatility of news about expected short term real rates. These conditionally log normal models generate news about expected excess returns only through shocks to conditional volatilities of macroeconomic shocks. The amount of news this mechanism produces depends on the average level of bond risk premia, which are much too small to allow for sizable volatilities of news about expected excess returns. Dynamic New Keynesian models also produce low volatilities of real rate news because the dynamics are not sufficiently persistent. Since these models do not have conditional log normal dynamics, nonlinearities can potentially create more news about expected excess 3

5 returns than can models in the tradition of Bansal and Yaron. But for parameterized models in the literature, the nonlinearities are not sufficient to generate realistic volatilities of shocks to yields. Habit formation preferences in the spirit of Campbell and Cochrane (1999) break the link between expected consumption growth and short term real rates, creating another mechanism for generating real rate news. In addition, the models nonlinearities can create substantial news about expected excess returns. Thus we can choose parameterizations of these models that are consistent with observed inflation variance ratios. However, the evidence is much too tentative to warrant the conclusion that nominal bond dynamics are best understood through the lense of habit formation. In particular, parameterizations that are successful in reproducing inflation variance ratios exhibit other properties that appear implausible. Empirically, innovations to expected short term real rates and expected excess returns are the primary drivers of yield shocks. Unfortunately, there is insufficient information in the data to disentangle the relative contributions of these two components at least without imposing restrictive assumptions. The relevant properties of the data are easy to summarize. Shocks to short term real rates are large, and long term nominal yields covary strongly with them. If short term real rates are highly persistent, then the variation in long term yields is explained by shocks to average expected future short term real rates. If short term real rates die out quickly, the variation is explained by shocks to expected excess returns also called shocks to term premia that positively covary with short term real rates. Point estimates of the persistence are consistent with the latter version, but statistical uncertainty in these estimates cannot rule out the former version. Section I describes how I measure inflation variance ratios and discusses the data used to construct shocks to inflation expectations. Section II documents the low level of the ratio 4

6 in the data. Section III discusses volatilities of components of yield shocks in various macro finance equilibrium models. Section IV attempts to determine the relative roles of news about expected future short rates and news about expected future returns. All references in the text to an Appendix are to an Internet Appendix, which contains detailed discussions of various issues. I. Inflation Variance Ratios Inflation news is not a clearcut concept. There is no unique or best way to measure how shocks associated with an inflation process affect nominal bond yields. The New Keynesian model examined by Rudebusch and Swanson (2012) helps illustrate the ambiguity. There are no exogenous shocks to inflation. Thus in a narrow sense there is no inflation news. However, shocks to productivity, monetary policy, and government spending each affect the paths of expected inflation, real rates, and nominal bond risk premia. Thus in a broad sense all news is inflation news, in the sense that every shock is information about expected future inflation. Models in which a monetary authority follows a Taylor rule typically have the same property. Outside of special cases, a shock to any variable that appears in the Taylor rule affects both yields and expected future inflation. Rather than adopt a specific model s interpretation of inflation shocks, I measure the magnitude of inflation news using an accounting approach that has its roots in the dividend/price decomposition of Campbell and Shiller (1988), as extended to returns by Campbell (1991). The measure is straightforward to estimate with available data. Any dynamic model of both inflation and bond yields a class of that includes a wide variety of dynamic macro models implies a value of a bond s inflation variance ratio. A. An Accounting Identity 5

7 I closely follow Campbell and Ammer (1993), who decompose unexpected bond returns into news about future real rates, news about future inflation, and news about future excess returns. The only mechanical difference is that I examine innovations in yields rather than innovations in returns. However, as Section II.C discusses, the conclusions I draw about the role of inflation contrast sharply with those of Campbell and Ammer. Begin with notation. All yields are continuously compounded and expressed per period. For example, with quarterly periods, a yield of 0.02 corresponds to 8% per year. y (m) t : Yield on a nominal zero coupon bond maturing at t + m. π t : Log change in the price level from t 1tot. r t : Ex ante real rate, the yield on a one period nominal bond less expected inflation, r t y (1) t E t (π t+1 ). Note that the ex ante real rate is not the rate on a one period real bond. In no-arbitrage, complete market models the ex ante real rate differs from the yield on a one period real bond owing to both a Jensen s inequality term associated with price shocks and the compensation investors require to face uncertainty in next period s price level. Investors that disagree about next period s expected inflation will also disagree about the level of the ex ante real rate. Discussion of inflation expectations is deferred until Section I.C. The log return to holding an m period nominal bond from t to t + 1 in excess of the log return to a one period nominal bond is ( ex (m) t+1 = my (m) t ) (m 1)y (m 1) t+1 y (1) t. (1) 6

8 An accounting identity decomposes the m maturity yield into future average inflation, ex ante real rates, and these excess log returns. It is y (m) t = 1 m m E t+i 1 (π t+i )+ 1 m i=1 m r t+i m i=1 m i=1 ex (m i+1) t+i. (2) The accounting identity formalizes observations such as holding constant a bond s yield, higher expectations of inflation over the life of the bond must correspond to either lower ex ante real rates or lower excess returns. Future expectations of inflation appear in (2) rather than realized inflation because the short rate in the excess return definition (1) is nominal rather than real. This identity holds regardless of the way in which inflation expectations are calculated. The time t expectation of (2) decomposes the m period yield into expectations of average inflation, average ex ante real rates, and average excess returns over the life of the bond. Using iterated expectations, the bond yield is y (m) t = 1 m m E t (π t+i )+ 1 m i=1 m E t (r t+i 1 )+ 1 m i=1 m ( E t i=1 ex (m i+1) t+i ). (3) Again, this equation is an identity regardless of the process for calculating expectations. The third sum on the right of (3) is often described as the bond s term premium. The accounting identity puts no structure on the term premium. In a frictionless no-arbitrage setting the term premium is determined by the risk premium investors require to hold the bond and a Jensen s inequality component associated with the log transformation. In models with frictions the term premium may also include a safety or convenience component. Using this accounting framework, express the innovation in the m maturity yield from t 1totasthe sum of news about expected average inflation, ex ante real rates, and excess 7

9 returns. Denote the news by y (m) t E t 1 y (m) t, ( ) ( ) 1 m 1 m π,t E t π t+i E t 1 π t+i, m m i=1 i=1 ( ) ( ) 1 m 1 m r,t E t r t+i 1 E t 1 r t+i 1, m m i=1 i=1 ( ) ( 1 m ex,t E t ex (m i+1) 1 m t+i E t 1 m m ỹ (m) t η (m) η (m) η (m) i=1 i=1 ex (m i+1) t+i ). (4) A yield shock is then the sum of news, or ỹ (m) t = η (m) π,t + η (m) r,t + η (m) ex,t. (5) This paper uses (5) to study the relative contributions of different types of news to yield innovations. The unconditional variance of yield innovations is the sum of the unconditional variances of the individual components on the right side of (5) and twice their unconditional covariances: ( Var ỹ (m) t ) ( =Var η (m) π,t +2Cov ) ( +Var ( η (m) π,t,η (m) r,t η (m) r,t ) ) +2Cov ( +Var η (m) ex,t ) ( η (m) π,t,η (m) ex,t ) ( ) +2Cov η (m) r,t,η (m) ex,t. (6) Divide (6) by the variance on the left to express the fraction of the variance explained, in an accounting sense, by news about expected inflation, expected real rates, and expected excess returns. I use the term inflation variance ratio to refer to one of these ratios, the variance of inflation news to the variance of yield shocks. The unconditional inflation variance ratio 8

10 is inflation variance ratio VR (m) π = ( Var ( Var η (m) π,t ỹ (m) t ) ). (7) It is important to understand what is, and what is not, measured by the inflation variance ratio. The magnitude of expected inflation news is not a summary measure of the difference in risk between nominal and real bonds because it does not capture all of the uncertainty investors bear when they bet on future inflation. For example, a shock to the conditional variance of inflation will alter the risk of nominal bonds and thereby change bond prices. In this accounting framework, such a shock appears in news about expected excess returns, not news about expected inflation. Also note that inflation news is not necessarily orthogonal to other news. For example, in the New Keynesian model discussed above, inflation news and real rate news are correlated. More broadly, the inflation variance ratio is not a fundamental measure of inflation risk derived from a macro finance model. It is better to view the inflation variance ratio as an informative moment of the data rather than a fundamental measure of inflation risk. Campbell (1991) is the obvious analogy, both formally and intuitively. Campbell s conclusion that news about future cash flows accounts for less than half the variation in aggregate stock returns strongly challenges macroeconomic models of equity prices. Similarly, the conclusions here are a strong empirical challenge to macroeconomic models of nominal yields. B. Conditional and Unconditional Ratios The terms conditional and unconditional can create some confusion because the shocks defined in (4) use conditioning information, while the variance ratio (7) does not. The focus here is on unconditional second moments of one step ahead shocks. Unconditional 9

11 variance ratios are ratios of average conditional variances, VR (m) π = ( E (Var t 1 ( E (Var t 1 η (m) π,t ỹ (m) t )) )). (8) (This equation uses the fact that conditional means of shocks are identically zero.) Sample inflation variance ratios are calculated using sample variances of one step ahead shocks. The sample variance ratios are then compared with corresponding ratios of unconditional variances implied by workhorse macro finance models. Conditional second moments can be used in (6) instead of unconditional second moments. For example, we could calculate inflation variance ratios conditioned on time t information. I do not focus on conditional variance ratios, although they are worth a detailed study. Rigorous analysis of conditional moments requires an explicit model of conditioning information. Balduzzi and Lan (2014) take a conditional approach to interpreting the news content of shocks to the ten year bond yield. Cram (2016), building on an earlier version of this research, models the dynamics of conditional inflation variance ratios using specific conditioning information. Here I get substantial mileage out of unconditional ratios without attempting to characterize conditional variances. I do, however, estimate sample inflation variance ratios for interesting subperiods. Subperiod results shed (model free) light on the time variation in conditional variance ratios, which in turn helps us evaluate the econonomic significance of the wedge between full sample variance ratios and model implied unconditional variance ratios. For example, if a model is incapable of matching full sample results, but is better able to match results from the 1970s and 1980s, we might conclude the model helps us understand a high inflation regime in the U.S. 10

12 Bauer and Rudebusch (2017) also extend the results of this research by modifying conditioning information. They generalize the shocks defined in (4) to h ahead shocks, while retaining the focus on unconditional second moments. As h gets large, the numerator of the inflation variance ratio converges to the unconditional variance of average expected inflation, while the denominator converges to the unconditional variance of the bond s yield. Cieslak and Povala (2015) discuss the long run relation between inflation expectations and bond yields. C. Measuring Innovations in Inflation Expectations Like many other researchers beginning with Pennacchi (1991), I infer inflation expectations from surveys of market practitioners. Consensus forecasts in other words, cross sectional means from these surveys are close in spirit to the subjective expectations of a sophisticated investor, although no agent s beliefs may correspond exactly to consensus forecasts. Substantial research concludes that forecasts from econometric models of inflation dynamics are not more accurate than consensus survey forecasts. Ang, Bekaert, and Wei (2007) document that survey forecasts are more accurate than model based forecasts constructed using the history of inflation and other non-survey information. In addition, they find no evidence that using realized inflation in addition to survey forecasts helps reduce survey based forecast errors. Faust and Wright (2009) and Croushore (2010) draw the same conclusion. Chernov and Mueller (2012) cannot reject the hypothesis that the subjective probability distribution of future inflation, as inferred from surveys, equals the true probability distribution. In a comprehensive handbook chapter, Faust and Wright (2013) concur:...purely judgmental forecasts of inflation are right at the frontier of our forecasting ability. 1 This earlier work supports the interpretation of consensus forecasts as expectations of 11

13 both market participants and researchers. (We may want to allow for measurement error, an issue that is discussed in the next section.) This research uses inflation forecasts from two types of Blue Chip (BC) surveys and the Survey of Professional Forecasters (SPF). The BC data are monthly beginning with March The SPF data are quarterly beginning in 1968Q4. The data samples used here run through The BC consensus forecasts are means across respondents. The SPF consensus forecast is the mean across the respondents, dropping outliers. 2 Survey forecasts are concentrated at relatively short horizons. The length of the cross section from BC surveys varies across observations, up to a maximum of seven quarters ahead. The SPF has forecasts for only four future quarters. Section II.B uses an econometric model to extend the the information in survey forecasts to longer horizon forecasts. D. Estimating Yield Innovations Shocks to bond yields as defined by (4) are realizations less the previous period s forecast. Survey forecasts of Treasury yields are available for a variety of maturities. Unlike inflation forecasts, survey forecasts of yields are not superior to or even as accurate as less subjective forecasts. Cieslak (2016) and Giacoletti, Laursen, and Singleton (2015) show that the martingale assumption produces forecasts that have lower root mean squared errors than consensus survey forecasts. Therefore the denominator of the inflation variance ratio (7) will be larger when evaluated using survey forecast than when using martingale forecasts. Since an important message of this paper is that the ratio (7) is quite small, I make the conservative choice to not use consensus survey forecasts of yields. Instead I use methods advocated in the empirical term structure literature. Research beginning with Duffee (2002) documents that martingale forecasts of Treasury bond yields typically have lower root mean squared errors in pseudo out of sample forecasting than do forecasts produced by 12

14 parameterized models. Thus the benchmark forecasts in this paper are martingale forecasts. I also explore using the shape of the short end of the yield curve to predict future changes in yields. The evidence of Campbell and Shiller (1991) supports this approach. In practice, as the results in the next section document, this choice does not have much of an effect on measures of inflation variance ratios. Yields are taken from two sources. The one quarter yield is from the Federal Reserve Board s H15 release. Yields on zero coupon bonds with maturities from two to six quarters, as well as five and ten years, are produced by Anh Le as described in Le and Singleton (2013). 3 I use both month end yields and mid month yields, depending on whether the yields are to be matched with BC forecasts or SPF forecasts. 4 All yields are continuously compounded and expressed at an annual rate. Measuring Inflation Variance Ratios This section estimates the inflation variance ratio measure (7) at both short and long horizons. Survey data allow the model free construction of news about average expected inflation over short horizons. Longer horizon forecasts require a dynamic model of inflation expectations. A simple model drawn from the literature on inflation expectations is used in Section II.B to estimate inflation variance ratios at multi year horizons. A. Short horizon forecasts A survey at quarter t reports k quarter ahead consensus predictions of inflation for k = 1,...,K max. Discard the one quarter ahead prediction, and convert the others to predictions of average log inflation from quarter t + 1 to quarter t + k, k =2,...,K max.use the survey at quarter t + 1 to calculate predictions of average log inflation over the same K max 1 horizons. The latter predictions minus the former predictions are the consensus 13

15 innovations at quarter t+1 of average expected inflation from t+1 to t+k, k =2,...,K max. No model is necessary to construct these measures of news about expected future inflation. 5 FIGURE 1 AROUND HERE Figure 1 displays time series of quarterly news about average expected inflation produced with consensus forecasts from BC surveys. The horizons range from one to six quarters. 6 A glance at the figure reveals that news is highly correlated across horizons. Part of the correlation is mechanical, since expected inflation over the next k quarters is embedded in average expected inflation over the next k + 1 quarters. Also note that the magnitude of news declines with the forecast horizon. For example, news about expected inflation over the next year is less volatile than news about expected inflation over the next quarter. Figure 2 displays similar news produced with consensus forecasts from SPF. The time series is longer and the cross section is shorter. The horizons range from one to three quarters. The long time series reveals substantial heteroskedasticity of news, with volatility peaking during the late 1970s. FIGURE 2 AROUND HERE I use two methods to construct corresponding yield innovations. One imposes the martingale assumption, so that current yields equal expected future yields. The other uses in sample forecasts produced by a regression that predicts future changes in short maturity yields using the shape of the short end of the yield curve. The regression equation is ( y (m) t+1 y (m) t = b 0,m + b 1,m y (1) t y (4) t y (6) t ) +ỹ (m) t+1. (9) In (9), both time and maturities are measured in quarters. The parameters b 0,m and b 1,m are 14

16 a scalar and a length-three vector respectively. In words, quarterly changes in bond yields are predicted using yields on one quarter, four quarter, and six quarter bonds. The residuals are the innovations. FIGURE 3 AROUND HERE Figure 3 displays quarterly innovations in yields from the regressions. The sample period and the maturity range match those for inflation expectation news displayed in Figure 1. Figure 4 also displays quarterly innovations, with a sample period and maturity range matching those in Figure 2. (The regressions are estimated separately for the two sample periods.) As with news about average inflation, yield innovations are highly correlated across horizons. One important difference is that volatilities of yield innovations decline only slightly with maturity. This difference implies that inflation variance ratios decline with maturity. FIGURE 4 AROUND HERE A glance at the scales of the vertical axes of these four figures reveals the main result of this paper. The typical magnitude of yield innovations is more than twice as large as the typical magnitude of corresponding news about average expected inflation. An immediate implication is that the ratio of the two variances is less than a quarter. We need to make some numerical calculations, extend the analysis to longer maturities, and evaluate statistical significance. But nothing that follows is surprising, given the evidence of Figures 1 through 4. A.1. Unconditional Inflation Variance Ratios Point estimates of unconditional inflation variance ratios are displayed in Table I. The Appendix describes how the robust standard errors are estimated with Generalized Method 15

17 of Moments (GMM). TABLE I AROUND HERE The table s main result is that unconditional inflation variance ratios are small, in the neighborhood of 0.1 to 0.2. Estimates in Panel A are for the 1980Q1 through 2013Q4 period, with inflation news from Blue Chip surveys. 7 Because of the sparseness of data for five quarter and six quarter horizons (see the gaps in Figure 1), results are displayed only for horizons up to four quarters. In Panel A the largest variance ratio for the one quarter yield is only 0.2. Estimates of variance ratios for three quarter and four quarter yields are less than 0.1. Estimates in Panel B are for the 1968Q4 through 2013Q4 period, with inflation news from SPF forecasts of the GDP deflator. The estimates are all less than a quarter, and decline with maturity. Standard errors for these variance ratios are sufficiently small that we can reliably conclude population variance ratios are less than a half for the one quarter yield and less than a quarter for maturities the at the longer maturities. Standard errors for the SPF estimates are slightly larger, but we can reliably conclude population variance ratios are less than 0.5 for the one quarter yield and less than 0.25 for the one year yield. The estimates in Table I assume that both inflation news and yield innovations are measured without error. This assumption is too strong. Consensus forecasts are cross sectional sample means, hence the period t forecast depends on the particular makeup of the panel at t. Bond yields are also measured with error. All but the one quarter yield are interpolated from yields on coupon bonds. Bekaert, Hodrick, and Marshall (1997) estimate that, for maturities around one year, standard deviations of interpolation measurement error are in the range of eight to nine basis points. 16

18 Measurement error is discussed in detail in the Appendix. Here it is sufficient to note that modest amounts of measurement error in both inflation news and yields will likely artificially increase observed inflation variance ratios as measured in Table I, but the effect will be negligible. A.2. Inflation Variance Ratios Across Monetary Regimes To reiterate, the evidence in Table I refers to unconditional inflation variance ratios. A glance at Figures 1 through 4 reveals considerable heteroskedasticity of both inflation news and yield shocks. Table II documents that conditional inflation variance ratios also vary through time. TABLE II AROUND HERE Results are presented for four subperiods. A common view is that monetary policy was accommodative during the pre-volcker years, ending in 1979Q2. Accommodation increases the responsiveness of expected inflation to macroeconomic shocks. In addition, as argued by Clarida, Galí, and Gertler (2000), it can produce sunspots in inflation expectations, creating another channel for news about expected inflation. The second period is disinflation, beginning in 1979Q3. Following Lubik and Schorfheide (2004) and others, I treat 1982Q2 as the end of the disinflation period. The aggressive monetary policy period begins with 1983Q1. The financial crisis/zero lower bound (ZLB) subperiod begins with the Lehman failure in 2008Q3 and continues through the end of the sample, 2013Q4. Before turning to detailed results, note that in Table I there is little difference between variance ratios calculated with regression based yield forecasts and martingale yield forecasts. In other words, quarterly changes in yields are largely unpredictable with the shape of the term structure. With subsamples there is a greater danger of overfitting. Therefore in Table 17

19 II all of the results use martingale yield forecasts. Three results in Table II are worth highlighting. First, outside of the crisis/zlb period, none of the reported inflation variance ratios exceeds a half. Second, they are higher during the period of accommodative monetary policy than during the period of aggressive monetary policy. Variance ratios during the former period are roughly 1.6 times corresponding variance ratios during the latter period. Third, extreme realizations of inflation variance ratios are observed during the extreme periods of the sample. Variance ratios are less than 0.1 during the disinflation period and well above a half during the crisis/zlb period. The discussion of these results is delayed until Section II.B, after estimation of inflation variance ratios for long maturity bonds. Table II also documents that holding the sample period fixed, variance ratios based on the SPF differ from those based on the Bue Chip surveys. The latter are typically larger than the former. The differences, which are not economically substantial, are briefly discussed in the Appendix. B. Longer Horizon Estimates Survey data on long horizon expectations of inflation are sparse. Blue Chip and SPF participants are occasionally asked to predict inflation at the five to ten year horizon, but neither the frequency of the responses not the precision of the inflation horizon allows for model free calculation of innovations to inflation expectations. Empirical implementation of (3) for bond maturities at horizons greater than a year requires a model. B.1. A Trend-cycle Model A conclusion of Faust and Wright (2013) motivates the modeling approach. They describe a method that produces intermediate range inflation forecasts close to the frontier 18

20 of predictive performance. Simply use a glide path to connect survey forecasts of current inflation to survey forecasts of distant inflation. A corollary to their conclusion, verified below, is that longer horizon forecasts can be extrapolated from the glide path on which short horizon forecasts lie. A trend-cycle model captures the intuition of the glide path approach. The model assumes a unit root in inflation, consistent with models of inflation such as Stock and Watson (2007) and Cogley, Primiceri and Sargent (2010). Period t inflation is the sum of three components. One follows a martingale, another follows a persistent stationary process, and the third is a serially uncorrelated shock. The equations are π t = τ t + ϕ t + φ t, (10) τ t = τ t 1 + ξ t, E t 1 (ξ t )=0, (11) ϕ t = θϕ t 1 + υ t, E t 1 (υ t )=0. (12) E t 1 (φ t )=0. (13) Following the discussion of Section I.D, long term bond yields are assumed to follow martingales, as in y (m) t = y (m) t 1 +ỹ (m) t, E t 1 ( y (m) t ) =0, (14) for some long maturity m. Similar models appear in Nason and Smith (2014), Stock and Watson (2007), and Cogley, Primiceri and Sargent (2010). More information about the model, including formulas for calculating inflation variance ratios, are in the Appendix. In this model both inflation expectations and long term yields have unit roots. This raises the question of cointegration. The earliest comprehensive empirical analysis of cointegration 19

21 among nominal yields is in Campbell and Shiller (1987). If both yields and inflation have unit roots, but they are not cointegrated, then either real rates or term premia must also have a unit root. Campbell and Ammer (1993) assume inflation and yields are cointegrated with a unit cointegrating vector. This assumption, imposed on the trend-cycle model, requires that the martingale component of inflation must move one for one with the long maturity yield (which is also a martingale). Hence inflation variance ratios must converge to one for long maturities. The Appendix has details. I do not impose cointegration, and allow the data to determine the magnitude of inflation variance ratios. This choice is consistent with results in the applied cointegration literature, which typically produces a double negative: we cannot reject the hypothesis that inflation and nominal yields are both nonstationary and not cointegrated. Examples include Lardic and Mignon (2004) and Hjalmarsson and Österholm (2010). In their critical review of the literature, Neely and Rapach (2008) conclude that...studies [of real rates] often report evidence of unit roots, or at a minimum substantial persistence. 8 Some readers may be uncomfortable with the assumption of unit roots in inflation and nominal yields. Section IV presents and estimates astationarymodel ofinflationand nominal yields. This choice has a minimal effect on measures of inflation variance ratios. B.2. Estimation Estimation uses current and expected future inflation from surveys, as well as observations of a five year bond yield. I then repeat the exercise using a ten year bond yield. 9 Since realized quarter t inflation is announced well after the end of quarter t (and after the bond yield is determined), I use survey consensus forecasts of current quarter inflation (nowcasts) rather than actual inflation. Survey consensus forecasts of future inflation are used up to the maximum available horizon. This maximum is seven quarters for the Blue Chip survey 20

22 and four quarters for the Survey of Professional Forecasters. With more observables than state variables, a stochastic singularity problem arises if variables are assumed to be measured without error. Therefore I assume all observables other than the inflation nowcast are contaminated by measurement error. There is nothing special about the nowcast s accuracy, but it is impossible to untangle measurement error from the purely transitory shock to inflation. The measurement error for a particular observable is assumed to be iid. Model parameters are estimated with exactly identified GMM. Robust standard errors adjust for conditional heteroskedasticity of shocks to yields and expected inflation. The Appendix contains more details about measurement error, estimation, and standard errors. B.3. Results Table III reports estimated inflation variance ratios. The Appendix contains all other estimates associated with the model. TABLE III AROUND HERE These long horizon results are consistent with the short horizon results in Tables I and II. For both the five year and ten year bonds, full sample point estimates of unconditional inflation variance ratios are all less than The standard errors allow us to reject at standard confidence levels the hypothesis that a variance ratio exceeds This conclusion is reinforced graphically in Figure 5. It displays, for the Blue Chip data, time series of the ten year bond yield and filtered estimates of expected inflation over the next ten years. The former series is much more volatile than the latter. Only a small part of the variation in bond yields is attributable to variations in expected future inflation over the life of the bond. FIGURE 5 AROUND HERE 21

23 It is important to verify that the long run inflation forecasts from this dynamic model are accurate, in the sense that they capture investor expectations of long run inflation. Visual evidence is in Figure 6. The circles are semiannual Blue Chip survey consensus forecasts of CPI inflation over the period beginning five years and ending ten years from the survey date. These data are unavailable prior to The x s are Blue Chip survey forecasts of GNP inflation over the same horizon. The solid line in Figure 6 displays filtered estimates from the trend-cycle model of expected inflation over the same future horizon. 10 The dashed line is explained in Section II.C. FIGURE 6 AROUND HERE A glance at the figure reveals that the Blue Chip and model implied forecasts closely correspond. In the early part of the sample, the GNP inflation survey forecasts are about 50 basis points lower than the model implied CPI forecasts. This is consistent with the mean difference between CPI inflation and GNP (and GDP) inflation. differences between the forecasts occur during the financial crisis. After 1985, the largest During the crisis the model s forecasts are modestly more volatile than are the survey forecasts. Table III also displays subsample results. The samples are the same as those for which short horizon inflation variance ratios are estimated in Table II. The joint results can be summarized as follows. Inflation variance ratios are highest during the passive monetary policy period of the 1970s. Yet even in this period, news about expected inflation accounts for less than half of the variation in yields. Inflation variance ratios are around one third for maturities greater than three months. During both the Volcker disinflation and the Great Moderation, inflation variance ratios are in the range of 0.05 to (Again, inflation variance ratiosare somewhat higher atthe three month horizon.) During the financial crisis/zlb 22

24 period, inflation variance ratios are very high well above a half at short maturities. But at long maturities, they are similar to those observed during both disinflation and the Great Moderation. C. Revisiting Campbell and Ammer The variance measure in Section I is borrowed from Campbell and Ammer s decomposition of excess bond returns. However, they conclude that shocks to average expected inflation account for the vast majority of shocks to nominal yields. Here I show that their conclusion relies on the assumption of cointegration, and produces strongly counterfactual estimates of expected long run inflation. They assume that expectations of future inflation and short term nominal rates are determined by the stationary dynamics of a vector autoregression (VAR). The six variables included in the VAR are the ex post real interest rate (short nominal rate at t less inflation at t + 1), the change in the short nominal rate, the excess return to the aggregate stock market, the slope of the term structure, the dividend-price ratio, and the relative bill rate. 11 Inflation enters only in the form of a linear combination with the short nominal rate, and the level of yields is not included. This setup implies that yields and inflation are nonstationary and cointegrated with a cointegrating vector of ones. In addition, since neither the level of inflation nor the level of some yield appears by itself in the VAR, these variables are implicitly assumed to not have any stationary components. By contrast, real rates are assumed to be stationary. Thus their assumptions differ sharply with the trend-cycle model of inflation in Section II.B. Campbell and Ammer produce estimates of inflation variance ratios for long term bonds that are substantially higher than those reported in Section II.B. They conclude that news about expected inflation accounts for effectively all of the variance of bond return shocks. To 23

25 confirm and update their results, I apply their methodology to more recent data. I use their data definitions and their monthly frequency. The sample is split after February 1987, which is the last month in the sample examined by Campbell and Ammer. Their methodology makes it easy to infer all of the components of (6), thus each is reported in Table IV. I do not bother calculating standard errors. TABLE IV AROUND HERE Table IV confirms the results of Campbell and Ammer, in the sense that estimated inflation variance ratios are about one for each sample. The full sample point estimate is about 1.3. The point estimates for the two subsamples are slightly less than one. Put differently, their VAR implies that the variance of news about expected average inflation over ten years is close to the variance of shocks to the ten year yield. When news about inflation is so large, expectations of long run inflation must vary substantially over time. Figure 6 displays with a dashed line the full sample monthly forecasts of average inflation from year five to year ten. The forecasts fluctuate substantially over time, from more than 12% in the early 1980s to less than 1% in It is clear from the figure that the VAR estimates of long run inflation are wildly at odds with both the survey forecasts and the forecasts produced by the models that allow some or all of the variability in inflation to be mean reverting. A straightforward implication of Figure 6 is that we should not impose the assumption of cointegration on the joint dynamics of inflation and yields. When we allow for more flexibility in their joint dynamics, estimated unconditional inflation variance ratios at long horizons are similar to model free estimates for short horizons. At both short and long horizons, the estimates are no more than

26 The results of this section naturally lead to two follow up questions that are addressed in the paper s next sections. First, how tightly do these results bind on our macroeconomic models? Second, since inflation news does not drive bond yields, what does? III. Volatilities in Standard Dynamic Models This section explains why standard long run risk models have substantial difficulty matching the volatilities of yield shocks and the volatilities of news about expected inflation documented in Section II. It also explains why habit formation models are less restrictive along the dimensions important for matching the documented behavior. A. A Recursive Utility Starting Point Bansal and Yaron (2004) combine recursive utility with long run consumption growth risk in an endowment economy. At first glance, it might seem that the evidence of Section II has no implications for models in the style of Bansal and Yaron. Their approach specifies the dynamics of the real economy without reference to an inflation process. Thus in principle, we can overlay any inflation process on a standard long run risk model; just pick one that matches the inflation variance ratios of Section II. However, the important modeling challenge posed by Section II is matching both the numerators and denominators of inflation variance ratios. Put differently, can a dynamic model produce both inflation variance ratios well below one and volatilities of yield shocks similar to those documented in Section II? Such a model must generate substantial news about either expected future short term real rates or expected future excess bond returns. The claim here is that long run risk models have difficulty generating much news of either type. The first extensive exploration of term structure dynamics in a long run risk framework 25

27 is Piazzesi and Schneider (2007). They add an exogeneous inflation process to a model with recursive utility and long run endowment risk. The elasticity of intertemporal substitution (EIS) is fixed at one. Their benchmark model is set in a log normal framework with homoskedastic shocks. Therefore there is no news about expected excess returns. Nominal yields react only to news about expected future short term real rates and news about expected future inflation. Standard deviations of these news components are readily calculated analytically. 12 The top panel of Figure 7 displays these model implied standard deviations. At all horizons, the magnitude of news about expected average inflation exceeds the magnitude of news about expected average real rates. Standard deviations of quarterly inflation news range from more than 60 basis points at the one quarter horizon to 30 basis points at the ten year horizon. Corresponding standard deviations for real rate news range from about 50 basis points at the short end to less than 7 basis points at the long end. FIGURE 7 AROUND HERE The parameterized model exhibits a standard property of long run risk models: low volatility of real rate shocks. Two key features of the long run risk approach are (a) shocks to expected consumption growth are small, and (b) the EIS is high. In combination, these features imply that short term real rates do not vary much over time. From quarter to quarter, news about expected future real rates is small. Thus shocks to nominal bonds must be driven primarily by shocks to expected inflation. In fact, Figure 7 shows that model implied standard deviations of yield shocks given bythesumofthetwotypesofnews arealmostallless than the corresponding standard deviations of news aboutexpected inflation. Real rate news and inflation news are sufficiently 26

28 negatively correlated that model implied inflation variance ratios are almost all greater than one. Negative news correlations are a consequence of macroeconomic dynamics that exhibit stagflation: negatively correlated shocks to expected consumption growth and expected inflation. Stagflation dynamics are consistent with the positive average risk premia earned by nominal bonds because nominal bonds unexpectedly decline in value when the future looks gloomy. Since real rates move in lockstep with expected consumption growth, news about real rates and news about expected inflation are also negatively correlated. Model implied news correlations range from about 0.3 at the one quarter horizon to 0.9 atthetenyear horizon. The same panel of Figure 7 also displays sample standard deviations of quarterly shocks to yields and expected inflation. They are for the 1968Q4 through 2013Q4 period, taken from Tables I and III. Model implied volatilities of inflation news are much larger than the sample values, while model implied volatilities of yield shocks are much smaller. A successful model requires smaller shocks to inflation expectations and much larger shocks to either real rates or expected excess returns. B. Adding Stochastic Volatility Bansal and Shaliastovich (2013) extend the approach of Piazzesi and Schneider (hereafter P/S) by including time varying conditional variances. This generalization opens the channel for shocks to expected excess returns, since shocks to conditional variances produce shocks to current and expected future risk premia. TABLE V AROUND HERE The top line of Table V summarizes a few important properties of the model estimated 27

29 by Bansal and Shaliastovich (hereafter B/S). The unconditional standard deviation of log differenced quarterly consumption is a little more than 1% (annualized). The correlation between consumption growth at quarters t and t+4 is The standard deviation matches, by construction, the sample standard deviation in postwar U.S. data. The fourth order serial correlation is a little high relative to the postwar U.S. value of 0.1. The model implied mean real yield curve slopes down because real bonds are a hedge. The nominal yield curve slopes up on average because stagflation risk outweighs the hedging properties of real bonds. 13 The bottom panel of Figure 7 displays model implied unconditional standard deviations of quarterly shocks, calculated using the point estimates of B/S. This panel plots one more function than does the P/S panel. In P/S, yield shocks are identical to the sum of news about expected real rates and expected inflation. In B/S, these differ because yield shocks include shocks to risk premia. The panel plots both the standard deviation of yield shocks and the standard deviation of the sum of news about expected real rates and news about expected inflation. The magnitude of news about expected excess returns determines the wedge between these standard deviations. The figure shows that the B/S model, like the P/S model, produces inflation variance ratios close to one. The B/S model shares with the P/S model the problematic features of news about expected real rates: there isn t much news, and the news is negatively correlated with news about expected future inflation. Perhaps surprisingly, there is also very little news about expected future excess returns. Standard deviations of yield shocks are almost identical to standard deviations of sums of news about expected real rates and news about expected inflation. At the ten year maturity these standard deviations differ by less than five basis points. Why is the time varying risk premium channel so small? The short answer is that quar- 28